Finiteness and Fluctuations in Growing Networks

TL;DR

This paper investigates how finite size and fluctuations influence the structure of growing networks, revealing unique finite-size effects and the Gaussian nature of degree fluctuations as networks grow large.

Contribution

It provides an exact degree distribution for finite networks using generating functions and analyzes the impact of initial conditions and fluctuations.

Findings

Finite networks exhibit unusual finite-size scaling behavior.

Degree fluctuations tend to Gaussian distribution as network size increases.

Fluctuations between network realizations are characterized by higher moments.

Abstract

We study the role of finiteness and fluctuations about average quantities for basic structural properties of growing networks. We first determine the exact degree distribution of finite networks by generating function approaches. The resulting distributions exhibit an unusual finite-size scaling behavior and they are also sensitive to the initial conditions. We argue that fluctuations in the number of nodes of degree k become Gaussian for fixed degree as the size of the network diverges. We also characterize the fluctuations between different realizations of the network in terms of higher moments of the degree distribution.